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Vol. 10 (2005), Paper no. 25, pages 865-900.
Journal URL
http://www.math.washington.edu/∼ejpecp/
Statistics of a Vortex Filament Model
Franco Flandoli and Massimiliano Gubinelli
Dipartimento di Matematica Applicata
Università di Pisa, via Bonanno 25 bis
56126 Pisa, Italy
Emails: f.flandoli@dma.unipi.it and m.gubinelli@dma.unipi.it
Abstract
A random incompressible velocity field in three dimensions composed by Poisson
distributed Brownian vortex filaments is constructed. The filaments have a random
thickness, length and intensity, governed by a measure γ. Under appropriate assumptions on γ we compute the scaling law of the structure function of the field and
show that, in particular, it allows for either K41-like scaling or multifractal scaling.
Keywords: Homogeneous turbulence, K41, vortex filaments, multifractal random
fields.
MSC (2000): 76F55; 60G60
Submitted to EJP on April 15, 2005. Final version accepted on June 1, 2005.
865
1
Introduction
Isotropic homogeneous turbulence is phenomenologically described by several theories,
which usually give us the scaling properties of moments of velocity increments. If u(x)
denotes the velocity field of the fluid and Sp (ε), the so called structure function, denotes
the p-moment of the velocity increment over a distance ε (often only its longitudinal
projection is considered), then one expect a behavior of the form
Sp (ε) = h|u(x + ε) − u(x)|p i ∼ εζp .
(1)
Here, in our notations, ε is not the dissipation energy, but just the spatial scale parameter
(see remark 5). Let us recall two major theories: the Kolmogorov-Obukov scaling law
(K41) (see [14]) says that
2
ζ2 =
3
probably the best result compared with experiments; however the heuristic basis of the
theory also implies ζp = p3 which is not in accordance with experiments. Intermittency
corrections seem to be important for larger p’s. A general theory which takes them into
account is the multifractal scaling theory of Parisi and Frisch [11], that gives us ζ p in the
form of a Fenchel-Legendre transform:
ζp = inf [hp + 3 − D(h)].
h∈I
This theory is a sort of container, which includes for instance the striking particular case
of She and Leveque [18]. We do not pretend to go further in the explanation of this topic
and address the reader to the monograph [9].
The foundations of these theories, in particular of the multifractal one, are usually
mathematically poor, based mainly on very good intuition and a suitable “mental image”
(see the beginning of Chapter 7 of [9]). Essentially, the scaling properties of S p (ε) are
given a priori, after an intuitive description of the mental image. The velocity field of the
fluid is not mathematically described or constructed, but some crucial aspects of it are
described only in plain words, and then Sp (ε) is given (or heuristically “deduced”).
We do not pretend to remedy here to this extremely difficult problem, which ultimately
should start from a Navier-Stokes type model and the analysis of its invariant measures.
The contribution of this paper is only to construct rigorously a random velocity field
which has two interesting properties: i) its realizations have a geometry inspired by the
pictures obtained by numerical simulations of turbulent fluids; ii) the asymptotic as ε → 0
of Sp (ε) can be explicitly computed and the multifractal model is recovered with a suitable
choice of the measures defining the random field. Its relation with Navier-Stokes models
and their invariant measures is obscure as well (a part from some vague conjectures, see
[7]), so it is just one small step beyond pure phenomenology of turbulence.
866
Concerning (i), the geometry of the field is that of a collection of vortex filaments,
as observed for instance by [19] and many others. The main proposal to model vortex
filaments by paths of stochastic processes came from A. Chorin, who made several considerations about their statistical mechanics, see [5]. The processes considered in [5] are
self-avoiding walks, hence discrete. Continuous processes like Brownian motion, geometrically more natural, have been considered by [10], [16], [6], [8], [17] and others. We do
not report here numerical results, but we have observed in simple simulations that the
vortex filaments of the present paper, with the tubular smoothening due to the parameter
` (see below), have a shape that reminds very strongly the simulations of [3].
Concerning (ii), we use stochastic analysis, properties of stochastic integrals and ideas
related to the theory of the Brownian sausage and occupation measure. We do not know
whether it is possible to reach so strict estimates on Sp (ε) as those proved here in the
case when stochastic processes are replaced by smooth curves. The power of stochastic
calculus seem to be important.
Ensembles of vortex structures with more stiff or artificial geometry have been considered recently by [1] and [12]. They do not stress the relation with multifractal models and
part of their results are numerical, but nevertheless they indicate that scaling laws can be
obtained by models based on many vortex structures. Probably a closer investigation of
simpler geometrical models like that ones, in spite of the less appealing geometry of the
objects, will be important to understand more about this approach to turbulence theory.
Let us also say that these authors introduced that models also for numerical purposes, so
the simplicity of the structures has other important motivations.
Finally, let us remark about a difference with respect to the idea presented in [5] and
also in [4]. There one put the attention on a single vortex and try to relate statistical
properties of the paths of the process, like Flory exponents of 3D self-avoiding walk, with
scaling law of the velocity field. Such an attempt is more intrinsic, in that it hopes to
associate turbulent scalings with relevant exponents known for processes. On the contrary,
here (and in [1] and [12]) we consider a fluid made of a multitude of vortex structures and
extract statistics from the collective behavior. In fact, it this first work on the subject,
we consider independent vortex structures only, having in mind the Gibbs couplings of
[8] as a second future step. Due to the independence, at the end again it is just the single
filament that determines, through the statistics of its parameters, the properties of S p (ε).
However, the interpretation of the results and the conditions on the parameters are more
in the spirit of the classical ideas of K41 and its variants, where one thinks to the 3D
space more or less filled in by eddies or other structures. It is less natural to interpret
multifractality, for instance, on a single filament (although certain numerical simulation
on the evolution of a single filament suggest that multifractality could arise on a single
filament by a non-uniform procedure of stretching and folding [2]).
867
1.1
Preliminary remarks on a single filament
The rigorous definitions will be given in section 2. Here we introduce less formally a few
objects related to a single vortex filament.
We consider a 3d-Brownian motion {Xt }t∈[0,T ] starting from a point X0 . This is the
backbone of the vortex filament whose vorticity field is given by
Z
U T
ρ` (x − Xt ) ◦ dXt .
(2)
ξsingle (x) = 2
` 0
The letter t, that sometimes we shall also call time, is not physical time but just the
parameter of the curve. All our random fields are time independent, in the spirit of
equilibrium statistical mechanics. We assume that ρ` (x) = ρ(x/`) for a radially symmetric
measurable bounded function ρ with compact support in the ball B(0, 1) (the unit ball in
3d Euclidean space). To have an idea, consider the case ρ = 1B(0,1) . Then ξsingle (x) = 0
outside an `-neighboor U` of the support of the curve {Xt }t∈[0,T ] . Inside U` , ξsingle (x) is a
time-average of the “directions” dXt , with the pre-factor U/`2 . More precisely, if x ∈ U` ,
one has to consider the time-set where Xt ∈ B(x, `) and average dXt on such time set.
The resulting field ξsingle (x) looks much less irregular than {Xt }t∈[0,T ] , with increasing
irregularity for smaller values of `.
When ρ is only measurable, it is not a priori clear that the Stratonovich integral
ξsingle (x) is well defined, since the quadratic variation corrector involves distributional
derivatives of ρ (the Itô integral is more easily defined, but it is not the natural object to
be considered, see the remarks in [6]). Since we shall never use explicitly ξsingle (x), this
question is secondary and we may consider ξsingle (x) just as a formal expression that we
introduce to motivate the subsequent definition of usingle (x). However, at least in some
particular case (ρ = 1B(0,1) ) or under a little additional assumptions, the Stratonovich
integral ξsingle (x) is well defined since the corrector has a meaning (in the case of ρ = 1B(0,1)
the corrector involves the local time of 3d Brownian motion on sferical surfaces).
The factor U/`2 in the definition of ξsingle (x) is obscure at this level. Formally, it could
be more natural just to introduce a parameter Γ, in place of U/`2 , to describe the intensity
of the vortex. However, we do not have a clear interpretation of Γ, a posteriori, from out
theorems, while on the contrary it will arise that the parameter U has the meaning of
a typical velocity intensity in the most active region of the filament. Thus the choice of
the expression U/`2 has been devised a posteriori. The final interpretation of the three
parameters is that T is the length of the filament, ` the thickness, U the typical velocity
around the core.
The velocity field u generated by ξ is given by the Biot-Savart relation
Z
U T
usingle (x) = 2
K` (x − Xt ) ∧ ◦dXt
(3)
` 0
868
where ∧ stands for the vector product in R3 , ◦ denote as usual Stratonovich integration
of semimartingales and the vector kernel K` (x) is defined as
Z
Z
1
x−y
1
ρ` (y)dy
dy,
V` (x) = −
.
(4)
K` (x) = ∇V` (x) =
ρ` (y)
3
4π B(0,`)
|x − y|
4π B(0,`) |x − y|
The scalar field V` satisfy the Poisson equation ∆V` = ρ` in all R3 . Since ρ is radially
symmetric and with compact support
R we can have two different situations according to
the fact that the integral of ρ: Q = B(0,1) ρ(x)dx is zero or not. If Q = 0 then the field
V` is identically zero outside the ball B(0, `). Otherwise the fields V` , K` outside the ball
B(0, `) have the form
`3
V` (x) = Q ,
|x|
K` (x) = −2Q`3
x
,
|x|3
|x| ≥ `.
(5)
Accordingly we will call the case Q = 0 short range and Q 6= 0 long range. The proof
of the previous formula for K is given, for completeness, in the remark at the end of the
section.
A basic result is that the Stratonovich integral in the expression of usingle can be
replaced by an Itô integral:
Lemma 1 Itô and Stratonovich integrals in the definition of usingle (x) coincide:
Z
U T
usingle (x) = 2
K` (x − Xt ) ∧ dXt
` 0
(6)
where the integral is understood in Itô sense. Then usingle is a (local)-martingale with
respect to the standard filtration of X.
About the proof, by an approximation procedure that we omit we may assume ρ
Hölder continuous, so the derivatives of K` exist and are Hölder continuous by classical
regularity theorems for elliptic equations. Under this regularity one may compute the
corrector and prove that it is equal to zero, so, a posteriori, equation (6) holds true in the
limit also for less regular ρ. About the proof that the corrector is zero, it can be done
component-wise, but it is more illuminating to write the following heuristic computation:
the corrector is formally given by
Z
1 T
−
(∇K` (x − Xt )dXt ) ∧ dXt .
2 0
Now, from the property dXti dXtj = δij dt one can verify that
(∇K` (x − Xt )dXt ) ∧ dXt = (curl K` ) (x − Xt )dt.
Since K` is a gradient, we have curl K` = 0, so the corrector is equal to zero.
869
Remark 1 One may verify that div usingle = 0, so usingle may be the velocity fluid of an
incompressible fluid. On the contrary, div ξsingle is different from zero and curl usingle is
not ξsingle but its projection on divergence free fields. Therefore, one should think of ξ single
as an auxiliary field we start from in the construction of the model.
Remark 2 Let us prove (5), limited to K to avoid repetitions. We want to solve ∆V = ρ
with ρ spherically symmetric. The gradient K = ∇V of V satisfies ∇ · K = ρ and, by
spherical symmetry, it must be such that
K(x) =
x
f (|x|)
|x|
for some scalar function f (r). By Gauss theorem we have
Z
Z
∇ · K(x)dx =
K(x) · dσ(x)
B(0,r)
∂B(0,r)
where dσ(x) is the outward surface element of the sphere. So
Z
2
f (r)4πr =
ρ(x)dx =: Q(r)
B(0,r)
namely f (r) =
Q(r)
.
4πr 2
Therefore
K(x) = Q(r)
x
4π|x|3
with Q(r) = Q(1) if r ≥ 1 (since ρ has support in B(0, 1)).
2
Poisson field of vortices
Intuitively, we want to describe a collection of infinitely many independent Brownian vortex filaments, uniformly distributed in space, with intensity-thickness-length parameters
(U, `, T ) distributed according to a measure γ. The total vorticity of the fluid is the sum
of the vorticity of the single filaments, so, by linearity of the relation vorticity-velocity,
the total velocity field will be the sum of the velocity fields of the single filaments.
The rigorous description requires some care, so we split it into a number of steps.
2.1
Underlying Poisson random measure
Let Ξ be the metric space
Ξ = {(U, `, T, X) ∈ R3+ × C([0, 1]; R3 ) : 0 < ` ≤
870
√
T ≤ 1}
with its Borel σ-field B (Ξ). Let (Ω, A, P ) be a probability space, with expectation denoted
by E, and let µω , ω ∈ Ω, be a Poisson random measure on B (Ξ), with intensity ν (a σfinite measure on B (Ξ)) given by
dν(U, `, T, X) = dγ(U, `, T )dW(X).
√
for γ a σ-finite measure on the Borel sets of {(U, `, T ) ∈ R3+ : 0 < ` ≤ T ≤ 1} and
dW(X) the σ-finite measure defined by
¸
Z
Z ·Z
ψ(X)dW(X) =
ψ(X)dWx0 (X) dx0
R3
C([0,1];R3 )
C([0,1];R3 )
for any integrable test function ψ : C([0, 1]; R3 ) → R where dWx0 (X) is the Wiener measure on C([0, 1], R3 ) starting at x0 and dx0 is the Lebesgue measure on R3 . Heuristically
the measure W describe a Brownian path starting from an uniformly distributed point in
all space. The assumptions on γ will be specified at due time.
The random measure µω is uniquely determined by its characteristic function
µ Z
¶
µ Z
¶
ϕ(ξ)
E exp i ϕ(ξ)µ(dξ) = exp − (e
− 1)ν(dξ)
Ξ
Ξ
for any bounded measurable function ϕ on Ξ with support in a set of finite ν-measure.
In particular, for example, the first two moments of µ read
Z
Z
E ϕ(ξ)µ(dξ) =
ϕ(ξ)ν(dξ)
Ξ
and
E
·Z
Ξ
ϕ(ξ)µ(dξ)
¸2
Ξ
=
·Z
ϕ(ξ)ν(dξ)
Ξ
¸2
+
Z
ϕ2 (ξ)ν(dξ).
Ξ
We have to deal also with moments of order p; some useful formulae will be now given.
2.2
Moments of the Poisson Random Field
Let ϕ : Ξ → R be a measurable function. We shall say it is µ-integrable if it is µ ω integrable for P -a.e. ω ∈ Ω. In such a case, by approximation by bounded measurable
compact support functions, one can show that the mapping ω 7→ µω (ϕ) is measurable.
If ϕ0 : Ξ → R is a measurable function with ν (ϕ 6= ϕ0 ) = 0, since P (µ (ϕ 6= ϕ0 ) = 0) =
1, we have that ϕ0 is µ-integrable and P (µ (ϕ) = µ (ϕ0 )) = 1. Therefore the concept of
µ-integrability and the random variable µ (ϕ) depend only on the equivalence class of ϕ.
Let ϕ be a measurable function on Ξ, possibly defined only ν-a.s. We shall say that it
is µ-integrable if some of its measurable extensions to the whole Ξ is µ-integrable. Neither
the condition of µ-integrability nor the equivalence class of µ (ϕ) depend on the extension,
by the previous observations. We need all these remarks in the sequel when we deal with
ϕ given by stochastic integrals.
871
Lemma 2 Let ϕ be a measurable function on Ξ (possibly defined only ν-a.s.), such that
ν(ϕp ) < ∞ for some even integer number p. Then ϕ is µ-integrable, µ(ϕ) ∈ Lp (Ω), and
E [µ (ϕ)p ] ≤ epp ν(ϕp ).
If in addition we have ν(ϕk ) = 0 for every odd k < p, then
E [µ (ϕ)p ] ≥ ν(ϕp ).
Proof. Assume for a moment that ϕ is bounded measurable and with support in a
set of finite ν-measure, so that the qualitative parts of the statement are obviously true.
Using the moment generating function
Eeλµ(ϕ) = eν(e
we obtain
∞
X
λp
p=0
p!
λϕ −1)
∞
X
¤n
1 £ λϕ
ν(e − 1)
n!
n=0
#n
"∞
∞
X
1 X λk
k
ν(ϕ )
=1+
n! k=1 k!
n=1
E[µ(ϕ)p ] =
=1+
∞
X
1 X λk1 +···+kn
ν(ϕk1 ) · · · ν(ϕkn )
n!
k
!
·
·
·
k
!
1
n
n=1
k ,...,k ≥1
1
=1+
∞
X
p=1
n
p
λ X
p! n=1
p
X
k1 ,...,kn ≥1
k1 +···+kn =p
p!
ν(ϕk1 ) · · · ν(ϕkn )
n!k1 ! · · · kn !
Hence we have an equation for the moments:
p
X
X
p!
p
E[µ(ϕ) ] =
ν(ϕk1 ) · · · ν(ϕkn )
n!k
!
·
·
·
k
!
1
n
n=1 k ,...,k ≥1
1
n
k1 +···+kn =p
Since ν(|ϕ|k ) ≤ [ν(|ϕ|p )]k/p for k ≤ p we have
E[|µ(ϕ)|p ] ≤ E[µ(|ϕ|)p ]
p
X
X
≤
n=1 k1 ,...,kn ≥1
k1 +···+kn =p
p
X
p
≤ ν(|ϕ| )
p!
ν(|ϕ|p )
n!k1 ! · · · kn !
X
n=1 k1 ,...,kn ≥0
k1 +···+kn =p
p
X np
p
p
= ν(|ϕ| )
n=1
n!
872
p!
n!k1 ! · · · kn !
≤ ep ν(|ϕ|p ).
(7)
This proves the first inequality of the lemma. A posteriori, we may use it to prove the
qualitative part of the first statement, by a simple approximation procedure for general
measurable ϕ.
For the lower bound, from the assumption that ν(ϕk ) = 0 for odd k < p, in the sum
(7) we have contributions only when all ki , i = 1, . . . , n are even. Then, neglecting many
terms, we have
E[µ(ϕ)p ] ≥ ν(ϕp ).
The proof is complete.
2.3
Velocity field
Let ρ be a radially symmetric measurable bounded function ρ on R3 with compact support
in the ball B(0, 1), and let K` be defined as in section 1.1. Then we have that
K1 is Lipschitz continuous.
(8)
Indeed by an explicit computation it is possible to show that
K1 (x) = −2Q(|x|)
with
Q(r) :=
Z
x
|x|3
ρ(x)dx
B(0,r)
(since ρ has support in B(0, 1) we have Q(r) = Q if r ≥ 1). Then
¸
·
x⊗x
x⊗x
1
0
∇K1 (x) = −2Q (|x|)
− 2Q(|x|)
−3
|x|4
|x|3
|x|5
with Q0 (r) = dQ(r)/dr and we can bound
|∇K1 (x)| ≤ Ckρk∞
since
Q(r) ≤ Ckρk∞ r3 ,
Q0 (r) ≤ Ckρk∞ r2 .
RT
For any x, x0 ∈ R3 , `, T > 0, the random variable X 7→ 0 K` (x − Xt ) ∧ dXt is
defined Wx0 -a.s. on C([0, 1], R3 ). We also have, given x ∈ R3 , `, T > 0, that X 7→
RT
K` (x − Xt ) ∧ dXt is a well defined measurable function, defined W-a.s. on C([0, 1], R3 ).
0
More globally, writing ξ = (U, `, T, X) for shortness, for any x ∈ R 3 we may consider the
measurable R3 -valued function
Z
U T
ξ
K` (x − Xt ) ∧ dXt
ξ 7→ usingle (x) := 2
` 0
873
defined ν-a.s. on Ξ. In plain words, this is the velocity field at point x of a filament
specified by ξ.
Again in plain words, given ω ∈ Ω, the point measure µω specifies the parameters and
locations of infinitely many filaments: formally
X
µ=
δξ α
(9)
α∈N
for a sequence of i.i.d. random points {ξ α } distributed in Ξ according to ν (this fact is
not rigorous since ν is only σ-finite, but it has a rigorous version by localization explained
below). Since the total velocity at a given point x ∈ R3 should be the sum of the
contributions from each single filament, i.e. in heuristic terms
X ξα
(10)
u(x) =
usingle (x)
α∈N
we see that, in the rigorous language of µ, we should write
Z
¡
¢
u(x) =
uξsingle (x) µ(dξ) = µ u·single (x) .
(11)
Ξ
¯p ¢
¡¯
If we show that ν ¯u·single (x)¯ < ∞ for some even p ≥ 2, then from lemma 2, u·single (x)
is µ-integrable and the random variable
¡
¢
ω 7→ u(x, ω) := µω u·single (x)
is well defined.
In some proof below we will use the occupation measure of the Brownian motion in
the interval [0, T ], which is defined, for every Borel set B of R3 as
B 7→
LTB
:=
Z
T
0
1Xt ∈B dt.
Lemma 3 Given x ∈ R3 and p > 0, there exist Cp > 0 such that for every `2 ≤ T ≤ 1we
have
£
¤
W W (x)p/2 ≤ Cp `p `T
where
W (x) =
Z
T
0
|K` (x − Xt )|2 dt.
Proof. Let us bound K by a multi-scale argument. This is necessary only in the longrange case (see the introduction). If |y| ≤ ` we can bound |K` (y)| ≤ C`. Indeed if |w| ≤ 1,
|K(w)| ≤ C for some constant C and then if |y| < ` we have |K` (y)| = `|K1 (y/`)| ≤ C`.
874
Next, given Λ > ` and an integer N , consider a sequence {`i }i=0,...,N of scales, with
` = `0 < `1 < ... < `N = Λ. Then, for i = 1, . . . , N , if `i−1 < |y| < `i , by the explicit
formula for K` (y) we have |K` (y)| ≤ C`(`/|y|)2 . Therefore
|K` (y)| ≤ C`
µ
`
`i−1
¶2
1`i−1 <|y|≤`i
If |y| > Λ we simply bound K`e (y) ≤ C`(`/Λ)2 .
Summing up,
!
Ã
N
X
|K` (y)|2 = |K` (y)|2 1|y|≤` +
1`i−1 <|y|<`i + 1|y|>Λ
i=1
2
≤ C` 1|y|≤` + C
N
X
i=1
`
2
µ
`
`i−1
¶4
1`i−1 <|y|<`i
µ ¶4
`
1|y|>Λ
+ C`
Λ
2
which implies the following bound for W (x):
W (x) ≤ C`
2
LTB(x,`)
+C
N
X
i=1
`
2
µ
`
`i−1
¶4
LTB(x,`i )\B(x,`i−1 )
µ ¶4
`
LTB(x,Λ)c
+ C`
Λ
2
where LTB has been defined above. By the additivity of B 7→ LTB , the sum appearing in
this equation can be rewritten as
"
¶4
¶4 µ ¶4 #
N
−1 µ
N µ
X
X
`
`
`
LTB(x,`i )\B(x,`i−1 ) =
−
LTB(x,`i )
`
`
`
i−1
i−1
i
i=1
i=1
¶4
µ
`
LTB(x,Λ) − LTB(x,`)
+
`N −1
"
¶4 µ ¶4 #
¶4
µ
N
−1 µ
X
`
`
`
T
≤
−
LTB(x,Λ) .
LB(x,`i ) +
`
`
`
i−1
i
N −1
i=1
Assume that `i /`i−1 ≤ 2 uniformly in i = 1, . . . , N . Then
¶4
N µ
X
`
LTB(x,`i )\B(x,`i−1 )
`
i−1
i=1
"
¶2 µ ¶2 # µ ¶2
µ ¶4
N
−1 µ
X
`
`
`
`
T
−
LB(x,`i ) + C
LTB(x,Λ) .
≤C
`i−1
`i
`i
Λ
i=1
875
Notice now that
N
−1
X
i=1
"µ
`
`i−1
¶2
µ ¶2 # µ ¶2 µ ¶2
`
`
`
=
−
−
≤1
`i
`
Λ
so that, by Cauchy-Schwartz and Jensen inequalities we have
"
#p/2
¶4
N µ
X
`
LTB(x,`i )\B(x,`i−1 )
`i−1
i=1
"
¶2 µ ¶2 # µ ¶p
µ ¶2p
N
−1 µ
X
¡ T
¢p/2
`
`
`
`
T
p/2
−
(LB(x,`i ) ) + Cp
LB(x,Λ)
.
≤ Cp
`i−1
`i
`i
Λ
i=1
An upper bound for W (x)p/2 is then obtained as
W (x)p/2 ≤ Cp `p (LTB(x,`) )p/2
µ ¶2p
µ ¶2p
¡ T
¢p/2
¡ T
¢p/2
`
`
p
p
LB(x,Λ)
LB(x,Λ)c
+ Cp `
+ Cp `
Λ
Λ
#µ ¶
"µ
¶
µ
¶
N
−1
2
2
p
X
`
`
`
p
−
(LTB(x,`i ) )p/2
+ C`
`
`
`
i−1
i
i
i=1
µ ¶2p
`
p
T
p/2
p
T p/2
≤ Cp ` (LB(x,`) ) + Cp `
Λ
"
¶2 µ ¶2 # µ ¶p
N
−1 µ
X
`
`
`
p
+ Cp `
−
(LTB(x,`i ) )p/2
`i−1
`i
`i
i=1
where we have used again Cauchy-Schwartz inequality.
We use now lemma 14 with α = 1. For a given λ ∈ (0, 1), we take both ε and ` equal
to λ in (24) and (25), and get
h¡
√
√
¢p/2 i
≤ Cp (λ ∧ T )p λ(λ ∨ T )2 .
W LTB(x,λ)
Then we obtain
£
¤
W W (x)p/2
µ ¶2p
`
T p/2
≤ Cp ` `T + Cp `
Λ
"
¶2 µ ¶2 # µ ¶p
N
−1 µ
X
√
√
`
`
`
−
(`i ∧ T )p `i (`i ∨ T )2
+ C p `p
`i−1
`i
`i
i=1
2p
p
876
and taking the limit as the partition gets finer:
¤
£
W W (x)p/2
µ ¶2p
`
2p
p
T p/2
≤ Cp ` `T + Cp `
Λ
" µ ¶#
Z Λ µ ¶p
2
√
√
`
`
p
2
p
(u ∧ T ) (u ∨ T ) ud −
+ Cp `
u
u
`
The integral can then be computed as
Z Λ µ ¶p
√
√
`
`2
(u ∧ T )p (u ∨ T )2 u 3 du
u
u
`
Z Λ
Z √T
du
= ` p `2 T
+ T p/2 `p+2 √ u−p du
2
u
T
`
√ −1
p 2
−1
−1 p/2 p+2
= ` ` T [` − T ] + (p − 1) T ` [T (1−p)/2 − Λ(1−p) ]
`
≤ `p `T + (p − 1)−1 `p √ `T
T
√
Using the fact that ` ≤ T and letting Λ → ∞ we finally obtain the claim.
¤
Remark 3 The multiscale argument above can be rewritten in continuum variables from
the very beggining by means of the following identity: if f : [0, ∞) → R is of class C 1 and
has a suitable decay at infinity, then
Z T
Z ∞
f (|x − Xt |) dt = −
f 0 (r) LTB(x,r) dr.
0
0
This identity can be applied to W (x). The proof along these lines is not essentially shorter
and perhaps it is more obscure, thus we have choosen the discrete multiscale argument
which has a neat geometrical interpretation.
Corollary 1 Assume
γ(U p `T ) < ∞
for some even integer p ≥ 2. Then, for any x ∈ R3 , we have
¯p ¢
¡¯
ν ¯u·single (x)¯ < ∞
and the random variable
has finite p-moment:
¡
¢
ω 7→ u(x, ω) := µω u·single (x)
E [|u(x)|p ] < ∞.
877
Proof. We have
¯p ¢
¡¯
ν ¯u·single (x)¯ =
Z
¯
¯p
Wx0 (¯u·single (x)¯ )dγ(U, `, T )dx0 .
By Burkholder-Davis-Gundy inequality, there is Cp > 0 such that
Hence
¯p
¯
¤
£
Up
Wx0 (¯u·single (x)¯ ) ≤ Cp 2p Wx0 W (x)p/2 .
`
¯
¯p
¤
Up £
W(¯u·single (x)¯ ) ≤ Cp 2p W W (x)p/2
`
p
U
≤ Cp0 2p `2p `T = Cp0 U p `T.
`
¯p ¢
¡¯ ·
Therefore ν ¯usingle (x)¯ < ∞ by the assumption γ(U p `T ) < ∞. The other claims are a
consequence of lemma 2.
¤
Lemma 4 Under the previous assumptions, the law of u(x, ·) is independent of x and is
invariant also under rotations:
L
Ru(x, ·) = u(Rx, ·)
for every rotation matrix R.
Proof. With the usual notation ξ = (U, `, T, X) we have
(U,`,T,X)
uξsingle (x) = usingle
(U,`,T,X−x)
(x) = usingle
xξ
(0) = uτsingle
(0)
where τx (U, `, T, X) = (U, `, T, X − x). The map τx is a measurable transformation of Ξ
into itself. One can see that ν is τx -invariant; we omit the details, but we just notice that
ν is not a finite measure, so the invariance means
Z
Z
ϕ (τx ξ) ν (dξ) =
ϕ (ξ) ν (dξ)
Ξ
Ξ
for every ϕ ∈ L1 (Ξ, ν). From this invariance it follows that the law of the random measure
µ is the same as the law of the random measure τx µ. Therefore
h
i
h
i
ξ
τx ξ
µ usingle (x) = µ usingle (0)
h
i
h
i
L
= (τx µ) uξsingle (0) = µ uξsingle (0) .
This proves the first claim.
878
If R is a rotation, from the explicit form of K` it is easy to see that
RK` (y) = K` (Ry)
hence
Ruξsingle
Z
U T
RK` (x − Xt ) ∧ dRXt
(x) = 2
` 0
Z
U T
K` (Rx − RXt ) ∧ dRXt
= 2
` 0
= uRξ
single (Rx)
L
where we have set R(U, `, T, X) = (U, `, T, RX). Again Rν = ν, Rµ = µ, so the end of
the proof is the same as above.
¤
We say that a random field u(x, ·) is homogeneous if its law is independent of x and
isotropic if its law is invariant under rotations.
Corollary 2 Assume
γ(U p `T ) < ∞
for every p > 1. Then {u(x, ·); x ∈ R3 } is an isotropic homogeneous random field, with
finite moments of all orders.
This corollary is sufficient to introduce the structure function and state the main results
of this paper. However, it is natural to ask whether the random field {u(x, ·); x ∈ R 3 }
has a continuous modification. Having in mind Kolmogorov regularity theorem, we need
good estimates of E [|u(x) − u(y)|p ]. They are as difficult as the careful estimates we shall
perform in the next section to understand the scaling of the structure function. Therefore
we anticipate the result without proof. It is a direct consequence of Theorem 1.
Proposition 1 Assume
γ(U p `T ) < ∞
for every p > 1. Then, for every even integer p there is a constant Cp > 0 such that
· µ
¶p ¸
` ∧ |x − y|
p
p
E [|u(x) − u(y)| ] ≤ Cp γ U
`T .
`
Consequently, if the measure γ has the property that for some even integer p and real
number α > 3 there is a constant Cp0 > 0 such that
· µ
¶p ¸
`∧ε
p
γ U
`T ≤ Cp0 εα for any ε ∈ (0, 1) ,
(12)
`
then the random field u(x) has a continuous modification.
879
Remark 4 A sufficient condition for (12) is: there are α > 3 and β > 0 such that for
every sufficiently large even integer p there is a constant Cp > 0 such that
γ [U p `T · 1`≤ε1−β ] ≤ Cp εα for any ε ∈ (0, 1) .
Indeed,
¶p ¸
`∧ε
`T
γ U
`
· µ
¸
¶p
¶p
¸
· µ
`∧ε
`∧ε
p
p
=γ U
`T · 1`≤ε1−β + γ U
`T · 1ε1−β ≤`
`
`
£
¤
≤ γ [U p `T · 1`≤ε1−β ] + γ U p εβp `T · 1ε1−β ≤`
·
p
µ
≤ Cp εα + εβp γ [U p `T ]
so we have (12) with a suitable choice of p. This happens in particular in the multifractal
example of section 3.2, remark 7.
The model presented here has a further symmetry which is not physically correct. This
symmetry, described in the next lemma, implies that the odd moments of the longitudinal
structure function vanish, in contradiction both with experiments and certain rigorous
results derived from the Navier-Stokes equation (see [9]). The same drawback is present
in other statistical models of vortex structures [1].
Beyond the rigorous formulation, the following property says that the random field
·
usingle has the same law as −u·single . We cannot use the concept of law since ξ does not
live on a probability space.
Lemma 5 Under the hypotesis of Corollary 1, given x1 , ..., xn ∈ R3 , the measurable vector
³
´
Un (ξ) := uξsingle (x1 ) , ..., uξsingle (xn )
has the property
Z
ϕ (Un (ξ)) dν (ξ) =
Z
ϕ (−Un (ξ)) dν (ξ)
for every ϕ = ϕ (u1 , ..., un ) : R3n → R with a polynomial bound in its variables..
Proof.
et = XT −t , we have
With the notation X
Z
U T
ξ
−usingle (xk ) = − 2
K` (xk − Xt ) ∧ ◦dXt
` 0
Z
U T
es ) ∧ ◦dX
es
= 2
K` (xk − X
` 0
= uSξ
single (xk )
e Since Sν = ν, we have the result (using the integrawhere S(U, `, T, X) = (U, `, T, X).
bility of Corollary 1).
¤
880
Lemma 6 If p is an odd positive integer, then
£­
®p ¤
ν u·single (y) − u·single (x) , y − x
=0
for every x, y ∈ R3 .
Proof.
It is sufficient to apply the lemma to the function
ϕ (u1 , u2 ) := hu1 − u2 , y − xip
and the points x1 = y, x2 = x, with the observation that
ϕ (−u1 , −u2 ) = −ϕ (u1 , u2 ) .
¤
2.4
Localization
At the technical level, we do not need to localize the σ-finite measures of the present work.
However, we give a few remarks on localization to help the intuitive interpretation of the
model. Essentially we are going to introduce rigorous analogs of the heuristic expressions
(9) and (10) written at the beginning of the previous section. The problem there was that
the law of ξ α should be ν, which is only a σ-finite measure. For this reason one has to
localize ν and µ.
Given A ∈ B (Ξ) with 0 < ν (A) < ∞, consider the measure µA defined as the
restriction of µ to A:
µA (B) = µ (A ∩ B)
for any B ∈ B (Ξ). It can be written (the equality is in law, or a.s. over a possibly
enlarged probability space) as the sum of independent random atoms each distributed
according to the probability measure B ∈ B (A) 7→ νeA (B) := ν(B|A):
µA (dξ) =
NA
X
δξα (dξ)
α=1
where NA is a Poisson random variable with intensity ν(A) and the family of random
variables {ξ α }α∈N is independent and identically distributed according to νeA . Moreover if
{Ai }i∈N is a family of mutually disjoint subsets of Ξ then the r.v.s {µAi }i∈N are independent.
Sets A as above with a physical significance are the following ones. Given 0 < η < 1
and R > 0, let
Aη,R = {(U, T, `, X) ∈ Ξ : ` > η, |x0 | ≤ R}.
881
In a fluid model we meet these sets if we consider only vortexes up to some scale η (it
could be the Kolmogorov dissipation scale) and roughly confined in a ball of radius R.
If we assume that the measure γ satisfies 0 < γ(` > η) < ∞ for each η > 0, then
0 < ν (Aη,R ) < ∞, and for the measure µη,R := µAη,R we have the representation
Nη,R
X
µη,R (dξ) =
δξα (dξ)
α=1
where Nη,R is P (ν(Aη,R )) and {ξ α }α∈N are i.i.d. with law νeη,R := νeAη,R .
For any x ∈ R3 we may consider ξ 7→ uξsingle (x) as a random variable in R3 , defined
νeη,R -a.s. on Aη,R . Moreover we may consider the r.v. uη,R (x) on (Ω, A, P ) defined as
Z
Z
ξ
uη,R (x) :=
usingle (x) µ(dξ) =
uξsingle (x) µη,R (dξ).
(13)
Aη,R
Ξ
It is the velocity field at point x, generated by the vortex filaments in Aη,R ∈ B (Ξ). In
this case we have the representation
Nη,R
uη,R (x) =
X
X Uα Z Tα
K` (x − Xtα ) ∧ dXtα
(x) =
α )2
(`
0
α=1
Nη,R
α
uξsingle
α=1
where the quadruples ξ α = (U α , `α , T α , X α ) are distributed according to νeη,R and are
independent. If ω 7→ ξ (ω) is any one of such quadruples, the random variable
ξ(ω)
ω 7→ usingle (x)
is well defined, since the law of ξ is νeη,R and the random variable ξ 7→ uξsingle (x) is well
defined νeη,R -a.s. on Aη,R . Therefore uη,R (x) is a well defined random variable on (Ω, A, P ).
We have noticed this in contrast to the fact that the definition of u(x) required difficult
estimates, because of the contribution of infinitely many filaments.
Given the Poisson random field, by localization we have constructed the velocity fields
uη,R (x) that have a reasonable intuitive interpretation. Connections between u η,R (x) and
u(x) can be established rigorously at various levels. We limit ourselves to the following
example of statement.
Lemma 7 Assume
for any p > 0. Then, at any x ∈ R3 ,
lim
γ(U p `T ) < ∞
(η,R−1 )→(0+ ,0+ )
E [|uη,R (x) − u(x)|p ] = 0.
882
Proof.
Since
uη,R (x) − u(x) =
Z
Ξ
we have
¡
¢
1Aη,R − 1 uξsingle (x) µ(dξ)
¯p ¤
£¯¡
¢
E [|uη,R (x) − u(x)|p ] ≤ Cp ν ¯ 1Aη,R − 1 u·single (x)¯
¯p ¤
£¡
¢¯
= Cp ν 1Aη,R − 1 ¯u·single (x)¯
¡
¢
Notice that we do not have ν Acη,R → 0, in general, so the argument to prove the lemma
must take into account the properties of the r.v. u·single (x). We have (by BurkholderDavis-Gundy inequality)
¯p ¤
£¡
¢¯
ν 1Aη,R − 1 ¯u·single (x)¯
Z
Z
¯
¯p
=
dx0
Wx0 (¯u·single (x)¯ )dγ(U, `, T )
|x0 |≥R
l≥η
Z
Z
£
¤
Up
p/2
W
(x)
dγ(U, `, T )
W
≤ Cp
dx0
x
0
2p
|x0 |≥R
l≥η `
Z p
Z
£
¤
U
p/2
≤ Cp
dγ(U,
`,
T
)
W
W
(x)
dx0
x
0
`2p
|x0 |≥R
where W (x) has been defined in lemma 3. Let us show that
Z
£
¤
Wx0 W (x)p/2 dx0 ≤ Cp `2p `T · θ (R)
(14)
|x0 |≥R
where θ (R) → 0 as R → ∞. The proof will be complete after this result. Recall from
the proof of lemma 3 that we have
µ ¶2p
`
p/2
p
p/2
p
T
W (x) ≤ Cp ` (LB(x,`) ) + Cp `
T p/2
Λ
"
¶2 µ ¶2 # µ ¶p
N
−1 µ
X
`
`
`
+ C p `p
−
(LTB(x,`i ) )p/2 .
`
`
`
i−1
i
i
i=1
From lemma 15 we have
Z
|x0 |≥R
W x0
≤ Cp (λ ∧
h¡
√
¢p/2
LTB(x,λ)
p
T ) λ(λ ∨
√
i
dx0
µ
R − (|x| + λ)
√
T ) exp −
T
2
883
¶
.
Hence
Z
|x0 | ≥R
¤
£
Wx0 W (x)p/2 dx0
¶
µ ¶2p
R − (|x| + `)
`
p
√
+ Cp `
T p/2
≤ Cp ` `T exp −
Λ
T
"
¶2 µ ¶2 # µ ¶p
N
−1 µ
X
`
`
`
−
+ C p `p
`i−1
`i
`i
i=1
µ
¶
√ p
√ 2
R − (|x| + `i )
√
× (`i ∧ T ) `i (`i ∨ T ) exp −
.
T
2p
µ
Repeating the arguments of lemma 3 we arrive at
Z
£
¤
Wx0 W (x)p/2 dx0
|x0 | ≥R
¶
µ ¶2p
µ
`
R − (|x| + Λ)
p
√
T p/2 .
+ Cp `
≤ Cp ` `T exp −
Λ
T
2p
Since T and ` are smaller than one, and p ≥ 2, we also have
Z
£
¤
Wx0 W (x)p/2 dx0
|x0 | ≥R
¡
¢
≤ Cp `2p `T exp (−R − (|x| + Λ)) + Λ−2p .
With the choice Λ = R/2 we prove (14). The proof is complete.
3
¤
The structure function
Given the random velocity field u (x, ·) constructed above, under the assumption of Corollary 2, a quantity of major interest in the theory of turbulence is the longitudinal structure
function defined for every integer p and ε > 0 as
Spk (ε) = E[he, u(x + εe) − u(x)i]p
(15)
where h·, ·i is the Euclidean scalar product in R3 , e ∈ R3 is a unit vector and x ∈ R3 , and
E, we recall, is the expectation on (Ω, A, P ).
Remark 5 We warn the reader familiar with the literature on statistical fluid mechanics
that ε here is not the dissipation energy, but just the spatial scale parameter. In the physical
literature, it is commonly denoted by `; however, in our mathematical analysis we need
two parameters: the scale parameter of the statistical observation, which we denote by ε,
and a parameter internal to the model that describes the thickness of the different vortex
filaments, that we denote by `.
884
k
The moments Sp (ε) depend only on ε and p, since u (x, ·) is homogeneous and isotropic:
if e = R · e1 where e1 is a given unit vector and R is a rotation matrix taking e1 on e,
using that the adjoint of R is R−1 , we have
E [hu(x + εe) − u(x), eip ] = E [hu(εe) − u(0), eip ]
= E [hRu(εe1 ) − u(0), eip ]
= E [hu(εe1 ) − u(0), e1 ip ] .
For this reason we do not write explicitly the dependence on x and e.
Let us also recall the (non-directional) structure function
Sp (ε) = E [|u(x + εe) − u(x)|p ]
k
which as Sp (ε) depends only on ε and p. We obviously have
Spk (ε) ≤ Sp (ε).
We shall see that, for even integers p, they have the same scaling properties. At the
technical level, due to the previous inequality, it will be sufficient to estimate carefully
k
Sp (ε) from below and Sp (ε) from above.
3.1
The main result
k
The quantities Sp (ε) and Sp (ε) describe the statistical behavior of the increments of the
velocity field when ε → 0 and have been extensively investigated, see [9]. Both are
expected to have a characteristic power-like behavior of the form (1). Our aim is to prove
that, for the model described in the previous section with a suitable choice of γ, (1) holds
true in the sense that the limit
log Sp (ε)
ζp = lim
(16)
ε→0
log ε
k
exists (similarly for Sp (ε)) and is computable. The following theorem gives us the necessary estimates from above and below, for a rather general measure γ. Then, in the
next subsection, we make a choice of γ in order to obtain the classical multifractal scaling
behaviour.
Theorem 1 Assume that
γ(U p `T ) < ∞
for every p > 1. Then, for any even integer p > 1 there exist two constants Cp , cp > 0
such that
· µ
¶p ¸
`∧ε
p
k
p
`T
(17)
cp γ [U `T 1`<ε ] ≤ Sp (ε) ≤ Sp (ε) ≤ Cp γ U
`
for every ε ∈ (0, 1).
885
The proof of this result is long and reported in a separate section.
We would like to give a very rough heuristic that could explain this result. It must be
said that we would not believe in this heuristic without the proof, since some steps are
too vague (we have devised this heuristic only a posteriori).
What we are going to explain is that
¶p
µ
`∧ε
p
p
`T.
W(|usingle (x + εe) − usingle (x)| ) ∼ U
`
This is the hard part of the estimate.
Let us discuss separately the case ε > ` from the opposite one. When ε > ` the
vortex structure usingle is very thin compared to the length ε of observation of the displacement, thus the difference usingle (x + εe)−usingle (x) does not really play a role and the
value of W(|usingle (x + εe) − usingle (x)|p ) comes roughly from the separate contributions
of usingle (x + εe) and usingle (x), which are similar. Let us compute W(|usingle (x)|p ).
Consider the expression (6) which defines usingle (x). Strictly speaking, consider the
short-range case, otherwise there is a correction which makes even more difficult the
intuition. Very roughly, K` (x − Xt ) behaves like ` · 1Xt ∈B(x,`) , hence, even more roughly,
usingle (x) behaves like
Z
U T
usingle (x) ∼
1Xt ∈B(x,`) dXt .
` 0
When Xt is a smooth curve, say a straight line (at distances compared to `), then the
RT
quantity 0 1Xt ∈B(x,`) dXt is roughly proportional to ` if Xt crosses B (x, `), while it is
zero otherwise. We assume the same result holds true when Xt is a Brownian motion. In
addition, Xt crosses B (x, `) with a probability proportional to the volume of the Wiener
sausage, which is `T . Summarizing, we have
½
Z T
` with probability `T
1Xt ∈B(x,`) dXt ∼
.
0
otherwise
0
Therefore usingle (x) takes rougly two values, U with probability `T and 0 otherwise. It
follows that W(|usingle (x)|p ) ∼ U p `T .
Consider now the case ε < `. The difference now is important. Since the gradient of
K` is of order one, we have
Z T
U
1Xt ∈B(x,`) dXt .
usingle (x + εe) − usingle (x) ∼ 2 ε
`
0
As above we conclude that usingle (x + εe) − usingle (x) takes rougly two¡ values,
εU/` with
¢p
probability `T and 0 otherwise. It follows that W(|usingle (x)|p ) ∼ U p ε` `T . The intuitive argument is complete.
886
3.2
Example: the multifractal model
The most elementary idea to introduce a measure γ on the parameters is to take U and T
as suitable powers of `, thus prescribing a relation between the thickness ` and the length
and intensity. That is, a relation of the form
dγ(U, `, T ) = δ`h (dU )δ`a (dT )`−b d`.
Moreover, we have to prescribe the distribution of ` itself, which could again be given by
a power law `−b d`. The K41 scaling described below is such an example.
Having in mind multi-scale phenomena related to intermittency, we consider a superposition of the previous scheme. Take a probability measure θ on an interval I ⊂ R+
(which measures the relative importance of the scaling exponent h ∈ I). Given two
functions a, b : I → R+ with a(h) ≤ 2 (to ensure `2 ≤ T ) consider the measure
Z
¤
£
dγ(U, `, T ) =
(18)
δ`h (dU )δ`a(h) (dT )`−b(h) d` θ(dh).
I
Then, according to Theorem 1 , we must evaluate
Z
p
γ(U `T 1`<ε ) =
`hp+1+a(h)−b(h) d`θ(dh)
Z[0,ε]×I
= cp,h εhp+2+a(h)−b(h) θ(dh)
(19)
I
while
¶p ¸ Z
· µ
Z
ε∧`
hp+1+a(h)−b(h)
p
p
`(h−1)p+1+a(h)−b(h) d`θ(dh)
`T =
`
d`θ(dh) + ε
γ U
`
[ε,+∞)×I
Z[0,ε]×I
= Cp,h εhp+2+a(h)−b(h) θ(dh)
I
As ε → 0, by Laplace method, we get
lim
ε→0
log Sp (ε)
= inf [hp + 3 − D(h)] = ζp
h∈I
log ε
with D(h) = b(h)−a(h)+1. With this choice of γ we have recovered the scaling properties
of the multifractal model of [11]. See [9] for a review.
Consider the specific choice θ(dh) = δ1/3 (dh) and a(1/3) = 2, b(1/3) = 4. We have
p
ζp = [hp + 3 − D(h)]h=1/3 = .
3
This is the Kolmogorov K41 scaling law for 3d turbulence. The choice a(1/3) = 2,
namely T = `2 , has the following geometrical meaning: the spatial displacement and
887
the thickness of the structure are comparable (remember that the curves are Brownian),
hence their shape is blob-like, as in the classical discussions of “eddies” around K41. The
choice b(1/3) = 4, namely the measure `−4 d` for the parameter `, corresponds to the
idea that the eddies are space-filling: it is easy to see that in a box of unit volume the
number of eddies of size larger that ` is of order `−3 . Finally, the choice θ(dh) = δ1/3 (dh),
namely U = `1/3 , is the key point that produces ζp = p3 ; one may attempt to justify it
by dimensional analysis or other means, but it is essentially one of the issues that should
require a better understanding.
Remark 6 K41 can be obtain from this model also for other choices of the functions a(h)
and b(h). For example, we may take θ(dh) = δ1/3 (dh), a(1/3) = 0 and b(1/3) = 2. The
value choice θ(dh) = δ1/3 (dh) is again the essential point. The value a(1/3) = 0 means
T = 1, hence the model is made of thin filaments of length comparable to the integral
scale, instead of blob-like objects. The value b(1/3) = 2, namely the measure ` −2 d` for
the parameter `, is again space filling in view of the major length of the single vortex
structures (recall also that the Hausdorff dimension of Brownian trajectories is 2). From
this example we see that in the model described here it is possible to reconcile K41 with a
geometry made of thin long vortexes.
Remark 7 Assume inf I > 0 and, for instance, supI D < ∞. Then limp→∞ ζp = +∞.
In particular, ζp > 3 for some even integer p. Since, by (19) and Laplace method,
γ(U p `T 1`<ε1−β ) ≤ Cp,β εζp (1−β)
for any β > 0, the condition of remark 4 is satisfied. Therefore the velocity field has a
continuous modification.
4
Proof of Theorem 1
Let us introduce some objects related to the structure functions at the level of a single
vortex filament. Let e be a given
the first element of the canonical basis, to
¡ · unit vector,
¢
fix the ideas. Since u(x) = µ usingle (x) , then
where
and similarly
hu(εe) − u(0), ei = µ [hδε usingle , ei]
δε usingle := u·single (εe) − u·single (0)
|u(εe) − u(0)| = |µ [δε usingle ]| .
Of major technical interest will be the quantities, of structure function type,
Spe (ε) = W(hδε usingle , eip )
They depend also on `, T, U .
Sp (ε) = W(|δε usingle |p ).
888
4.1
Lower bound
As a direct consequence of lemma 6 and lemma 2 we have:
Corollary 3 If k is an odd positive integer, then
k
Sk (ε) = 0.
Moreover, for any even integer p > 1 there is a constant cp such that
Spk (ε) ≥ cp ν [hδε usingle , eip ]
£
¤
= cp γ Spe (ε) .
If we prove that
Spe (ε) ≥ c0p U p `T
for every ε ∈ (`, 1) and some constant c0p > 0, then
£
¤
γ Spe (ε) ≥ c00p γ [U p `T 1`<ε ]
for every ε ∈ (0, 1) and some constant c00p > 0, which implies the lower bound of theorem
1. We have
Z
e
Sp (ε) = Wx0 [hδε usingle , eip ] dx0 .
Since
U
hδε usingle , ei = 2
`
where
Z
T
0
hK`e (εe − Xt ) − K`e (0 − Xt ), dXt i
K`e (y) = K` (y) ∧ e
by Burkholder-Davis-Gundy inequality, there is cp > 0 such that
i
h
Up
p
e p/2
Wx0 [hδε usingle , ei ] ≥ cp 2p Wx0 (Wε )
`
where
Wεe
Here
K`e (y)
=
Z
T
0
dt|K`e (εe − Xt ) − K`e (0 − Xt )|2 .
= hK` (y), ei. Therefore
Spe (ε)
Z
h
i
Up
e p/2
≥ cp 2p Wx0 (Wε )
dx0
`
i
Up h
= cp 2p W (Wεe )p/2 .
`
The proof of the theorem is then complete with the following basic estimate.
889
Lemma 8 Given p > 0, there exist cp > 0 such that for every `2 ≤ T ≤ 1 and
ε>`
we have
Proof.
Recall that
i
h
W (Wεe )p/2 ≥ cp `2p `T.
K1 (x) = −2Q
x
|x|3
for |x| ≥1.
Consider the function
f (z, α) = K1e (z) − K1e (z + αe)
= (K1 (z) − K1 (z + αe)) ∧ e
defined for z ∈ R3 and α ≥ 1. Let e⊥ be any unit vector orthogonal to e. We have
¯
¯
¡ ⊥
¢
¯
¯
¯ ¡ ⊥ ¢¯
e
+
αe
∧
e
¯
¯f e , α ¯ = 2Q ¯¯1 −
3 ¯¯
⊥
¯
|e + αe|
¯
¯
¯
¯
1
¯
= 2Q ¯¯1 −
3
|e⊥ + αe| ¯
¡
¢
≥ f e⊥ , 1 = C 0 Q
³
´
3
for C0 = 2 1 − 2− 2 . Moreover, K1 is globally Lipschitz, hence there is L > 0 such that
¯ ¡ ⊥
¢
¡
¢¯
¯f e + w, α − f e⊥ , α ¯
¯
¯ ¯
¯
≤ ¯K1 (e⊥ + w) − K1 (e⊥ )¯ + ¯K1 (e⊥ + αe + w) − K1 (e⊥ + αe)¯ ≤ L |w|
for every w ∈ R3 and α ≥ 1. Therefore, there exists a ball B(e⊥ , a) ⊂ B(0, 2) and a
constant C1 > 0 such that when z ∈ B(e⊥ , a) we have
|K1e (z) − K1e (z + αe)| ≥ C1
uniformly in α ≥ 1. Then reintroducing the scaling factor ` we obtain that for y ∈
B(`e⊥ , `a)
|K`e (y) − K`e (y + εe)| > C1 `
uniformly in ` ∈ (0, 1) and ε > `. Then we have
|K`e (0 − Xt ) − K`e (εe − Xt )| ≥ C1 `1Xt ∈B(−`e⊥ ,`a) .
890
Hence, if ε > ` we have
Wεe
=
Z
T
0
dt|K`e (0 − Xt ) − K`e (εe − Xt )|2
≥ C1 `2 LTB(−`e⊥ ,`a) .
From the lower bound in (24) proved in the next section,
·³
i
h
´p/2 ¸
p
e p/2
T
≥ cp ` W LB(−`e⊥ ,`a)
≥ c0p `2p `T.
W (Wε )
The claim is proved.
¤
Remark 8 With a bit more of effort is is also possible to prove the bound
µ
¶p
ε∧`
p
e
Sp (ε) ≥ cp U
`T
`
valid for every ε ∈ (0, 1) (not only for ε > `). This would be the same as the upper bound,
but we do not need it to prove that the behaviors as ε → 0 of the upper and lower bound
is the same.
4.2
Upper bound
Lemma 9 For every even p there exists a constant Cp > 0 such that
Sp (ε) ≤ Cp ν [|δε usingle |p ]
= Cp γ [Sp (ε)] .
Proof.
Let [δε usingle (ξ)]i be the i-th component of δε usingle (ξ). We have
Sp (ε) ≤ Cp
3
X
E
i=1
£¡ £
¤¢p ¤
µ [δε usingle (ξ)]i
and thus, by lemma 2,
Sp (ε) ≤ Cp0
3
X
£
¤
ν [δε usingle (ξ)]pi
i=1
which implies the claim.
It is then sufficient to prove the bound
Sp (ε) ≤ Cp U
p
µ
891
ε∧`
`
¶p
`T.
¤
Again as above, We have
Sp (ε) =
Z
Wx0 [|δε usingle |p ] dx0
where, by Burkholder-Davis-Gundy inequality, there is Cp > 0 such that
£
¤
Up
Wx0 [|δε usingle | ] ≤ Cp 2p Wx0 Wεp/2
`
p
where
Wε =
Therefore
Z
T
0
dt|K` (εe − Xt ) − K` (0 − Xt )|2 .
Z
£
¤
Up
Sp (ε) ≤ Cp 2p Wx0 Wεp/2 dx0
`
U p £ p/2 ¤
= Cp 2p W Wε .
`
It is then sufficient to prove the bound
W
£
Wεp/2
¤
≤ Cp `
2p
µ
ε∧`
`
¶p
`T.
For ε > ` it is not necessary to keep into account the closeness of εe to 0: each term
in the difference of Wε has already the necessary scaling. The hard part of the work has
been done in lemma 3 above.
Lemma 10 Given p > 0, there exist Cp > 0 such that for every `2 ≤ T ≤ 1 and
ε>`
we have
Proof.
Since
RT
¤
£
W Wεp/2 ≤ Cp `2p `T.
Wε ≤ 2W (εe) + 2W (0)
|K` (x − Xt )|2 dt, by lemma 3 we have the result.
¤
h
i
p/2
For ε ≤ ` need to extract a power of ε from the estimate of W Wε . We essentially
repeat the multi-scale argument in the proof of lemma 3, with suitable modifications.
where W (x) =
0
892
Lemma 11 As in the previous lemma, when
ε≤`
we have
Proof.
£
¤
W Wεp/2 ≤ Cp εp `p `T.
Since now ε is smaller that ` we bound K` (y) − K` (z) for |y − z| ≤ ε as
|K` (y) − K` (z)| ≤ Cε
if |y| ≤ 2`. If |y| > 2` then |z| ≥ ` and using the explicit form of the kernel K` we have
the bound
µ ¶2
µ ¶3
`
`
≤ Cε
|K` (y) − K` (z)| ≤ Cε
u
u
where u is the minimum between |y| and |z| and in this case `/u < 1. Then, given a
partition of [2`, Λ], say 2` = `0 < `1 < ... < `N = Λ, as in the proof of lemma 3, we get
Ã
!
N
X
2
2
|K` (y) − K` (z)| = |K` (y) − K` (z)| 1|y|≤2` +
1`i−1 <|y|<`i + 1|y|>Λ
2
≤ Cε 1|y|≤2` + C
2
4
≤ Cε 1|y|≤2` + 2 C
N
X
ε
N
X
2
i=1
ε
2
i=1
µ
`
`i−1
µ
i=1
`
`i−1 − `
¶4
¶4
1`i−1 <|y|<`i
1`i−1 <|y|<`i
µ ¶4
`
+ Cε
1|y|>Λ
Λ
2
µ ¶4
`
1|y|>Λ
+ Cε
Λ
2
where we have used the fact that (u − `)−1 ≤ 2/u for u ≥ 2`. Then
Wε ≤ Cε
2
LTB(0,2`)
+C
N
X
i=1
ε
2
µ
`
`i−1
¶4
LTB(0,`i )\B(x,`i−1 )
µ ¶4
`
LTB(x,Λ)c
+ Cε
Λ
2
and arguing as in that proof
Wε
p/2
µ ¶2p
µ ¶2p
¢p/2
¡ T
`
`
T
p/2
p
LB(x,Λ)c
≤ Cε
+ Cε
[LB(0,Λ) ] + Cε
Λ
Λ
#µ ¶
"µ
¶
µ
¶
N
−1
2
2
p
X
`
`
`
+ Cεp
−
(LTB(0,`i ) )p/2 .
`
`
`
i−1
i
i
i=1
p
(LTB(0,2`) )p/2
p
Then, from lemma 14 in the form
h¡
√
√
¢p/2 i
T
≤ Cp (λ ∧ T )p λ(λ ∨ T )2 .
W LB(x,λ)
893
and the obvious bound LTB ≤ T for every Borel set B, we have
W
+Cεp
N
−1
X
i=1
£
"µ
Wεp/2
`
`i−1
¤
¶2
µ ¶2p
`
T p/2
≤ Cε ` `T + Cε
Λ
p p
p
µ ¶2 # µ ¶p
√
√
`
`
−
(`i ∧ T )p `i (`i ∨ T )2
`i
`i
and taking the limit as the partition gets finer:
µ ¶2p
`
T p/2
≤ Cε ` `T + Cε
W
Λ
" µ ¶#
Z Λ µ ¶p
2
√
√
`
`
.
+Cεp
(u ∧ T )p (u ∨ T )2 ud −
u
u
2`
£
Wεp/2
¤
p p
p
A direct computation of the integral as in the proof of lemma 3 and the limit as Λ → ∞
complete the proof.
¤
5
Auxiliary results on Brownian occupation measure
In this section we prove the estimates on W[|LTB(u,`) |p/2 ] which constitute the technical
core of the previous sections. The literature on Brownian occupation measure is wide, so
it is possible that results proved here are given somewhere or may be deduced from known
results. However, we have not found the uniform estimates we needed, so we prefer to give
full self-contained proofs for completeness. Of course, several ideas we use are inspired by
the existing literature (in particular, a main source of inspiration has been [15]).
First, notice that W[|LTB(u,`) |p/2 ] does not depend on u. But this is not of great help.
When p = 2, W[|LTB(u,`) |p/2 ] can be explicitly computed:
W[LTB(u,`) ]
=
Z
dx0
R3
=
Z
= |B(u, `)|
Z
dx0
R3
T
dt
0
Z
T
Z
dt
0
Z
T
dt
0
Z
B(u,`)
dz pt (z − x0 )
dz pt (x0 )
B(u,`)
Z
pt (x0 ) dx0 = $`3 T
R3
where $ is a geometrical constant and pt (x) is the density of the 3D Brownian motion
at time t. The estimate of W[|LTB(u,`) |p/2 ] for general p requires much more work.
894
Let
τB(u,`) = inf{t ≥ 0 : Xt ∈ B(u, `)}
the entrance time in B(u, `) for the canonical process. We continue to denote by Wx0 the
Wiener measure starting at x0 and also the mean value with respect to it; similarly for
W, the σ-finite measure dWx0 dx0 .
Lemma 12 For any p > 0, T > 0, ` > 0, x0 , u ∈ R3 , it holds that
T /2
Wx0 [τB(u,`/2) ≤ T /2]W0 [|LB(0,`/2) |p ] ≤ Wx0 [|LTB(u,`) |p ]
≤ Wx0 [τB(u,`) ≤ T ]W0 [|LTB(0,2`) |p ].
(20)
Proof. Let us prove the upper bound. Set, for simplicity, τ = τB(u,`) ∧ T . When
τ ≤ t < T we have
Xt ∈ B(u, `) ⇒ Xt − Xτ ∈ B(0, 2`)
then
LTB(u,`)
=
Z
T
τ
1Xt ∈B(u,`) dt ≤
Z
T
τ
1{(Xt −Xτ )∈B(0,2`)} dt =
Z
T −τ
0
1(Xτ +t −Xτ )∈B(0,2`) dt
Then, taking into account that τ = T implies LTB(u,`) = 0,
LTB(u,`)
≤ 1{τ <T }
Z
T
0
1(Xτ +t −Xτ )∈B(0,2`) dt
which gives us, using the strong Markov property,
µZ T
·
¶p ¸
T
p
Wx0 |LB(u,`) | ≤ Wx0 1τ <T
1(Xτ +t −Xτ )∈B(0,2`) dt
0
¶p ¸
·µZ T
= Wx0 [τ < T ]W0
1Xt ∈B(0,2`) dt
0
£
¤
= Wx0 [τB(u,`) < T ]W0 (LTB(0,2`) )p .
The upper bound is proved.
Let us proceed with the lower bound. Let τ 0 = τB(u,`/2) ∧ T . When τ 0 ≤ t ≤ T we have
Xt − Xτ 0 ∈ B(0, `/2) ⇒ Xt ∈ B(u, `)
895
then
LTB(u,`)
≥
Z
T
1Xt ∈B(u,`) dt
τ0
Z T −τ 0
≥
Z
T
τ0
1Xt −Xτ 0 ∈B(0,`/2) dt
1Xτ 0 +t −Xτ 0 ∈B(0,`/2) dt
Z T −τ 0
0
1Xτ 0 +t −Xτ 0 ∈B(0,`/2) dt
≥ 1τ ≤T /2
0
Z T /2
1Xτ 0 +t −Xτ 0 ∈B(0,`/2) dt.
≥ 1τ 0 ≤T /2
≥
0
0
Then, using again the strong Markov property with respect to the stopping time τ 0 , we
obtain
T /2
Wx0 [|LTB(u,`) |p ] ≥ Wx0 [τB(u,`/2) ≤ T /2]W0 [(LB(0,`/2) )p ]
and the proof is complete.
¤
Letting p = 1 in the previous lemma and using the scale invariance of BM we obtain
an upper bounds for Wx0 [τB(u,`) ≤ T ] as
√
Wx0 [τB(u,`) ≤ T ] = Wx0 / 2 [τB(u/
√
√
2,`/ 2)
≤ T /2] ≤
Wx0 /√2 [LTB(u/√2,√2`) ]
T /2
W0 [LB(0,`/√2) ]
and the corresponding lower bound for Wx0 [τB(u,`/2) ≤ T /2]:
Wx0 [τB(u,`/2) ≤ T /2] =
W√
√
√
2x0 [τB( 2u,`/ 2)
≤ T] ≥
W√2x0 [LTB(√2u,`/√2) ]
W0 [LTB(0,√2`) ]
which leads to the following easy corollary:
Corollary 4 For any p > 0, T > 0, ` > 0, x0 , u ∈ R3 , it holds that
T /2
W√2x0 [LTB(√2u,`/√2) ]W0 [|LB(0,`/2) |p ]
T /2
W0 [LB(0,√2`) ]
≤ Wx0 [|LTB(u,`) |p ]
≤
896
Wx0 /√2 [LTB(u/√2,√2`) ]W0 [|LTB(0,2`) |p ]
W0 [LTB(0,`/√2) ]
(21)
Lemma 13 Given α > 0 and p > 0, there exist constants c, C > 0 such that the following
properties hold true: for every T, ` satisfying T /`2 ≥ α we have
c`p ≤ W0 |LTB(0,`) |p/2 ≤ C`p
while if T /`2 ≤ α
Proof.
cT p/2 ≤ W0 |LTB(0,`) |p/2 ≤ CT p/2 .
Consider first T /`2 ≥ α. In distribution
L
LTB(0,`) =
`
2
Z
T /`2
0
1Xt ∈B(0,1) dt
(22)
and moreover we have that
c ≤ W0
"Z
T /`2
0
1Xt ∈B(0,1) dt
#p
≤C
(23)
uniformly in T /`2 ≥ α > 0 (the constants depend on α and p). The lower bound is
obtained by setting T /`2 = α while the upper bound is given by the fact that
·Z ∞
¸p
W0
1Xt ∈B(0,1) dt < ∞
0
for any p > 0 (see for instance [13], section 3). From (22) and (23) we get the first claim
of the lemma.
Next, if T /`2 ≤ α, in distribution:
Z 1
L
T
LB(0,`) = T
1Xt ∈B(0,`/√T ) dt
0
and
W0 |L1B(0,1/√α) |p ≤ W0 |L1B(0,`/√T ) |p ≤ 1
so the second claim is also proved.
¤
Lemma 14 Given p > 0, α > 0, there are constants c, C > 0 such that, for T ≥ α`2
while for T ≤ αε2
c`p `T ≤ W[|LTB(u,`) |p/2 ] ≤ C`p `T
(24)
cT p/2 ε3 ≤ W[|LTB(u,ε) |p/2 ] ≤ CT p/2 ε3 .
(25)
897
Proof. Let us prove (24). Using lemma 13, equation (21) becomes
c`p−2 W√2x0 [LTB(√2u,`/√2) ] ≤ Wx0 [|LTB(u,`) |p/2 ] ≤ C`p−2 Wx0 /√2 [LTB(u/√2,√2`) ]
(26)
for two constants c, C > 0 (depending on α and p). Moreover, as we remarked at the
beginning of the section,
W[LTB(u,`) ] = $`3 T.
Using this identity in (26), we get (24).
Now consider eq.(25). Assume T ≤ αε2 . Using eq.(21) and lemma 13 we have
p
p
cT 2 −1 W√2x0 [LTB(√2u,ε/√2) ] ≤ Wx0 [|LTB(u,ε) |p/2 ] ≤ CT 2 −1 Wx0 /√2 [LTB(u/√2,√2ε) ]
(27)
and (25) is a consequence of the identity
W[LTB(u,ε) ] = $ε3 T.
The proof is complete.
¤
The previous lemma solves the main problem posed at the beginning of the section.
We prove also a related result in finite volume that we need for the complementary results
of section 2.4. We limit ourselves to the upper bounds, for shortness.
Lemma 15 Given p > 0, α > 0, u ∈ R3 , there is a constant C > 0 and a function θ (R)
with limR→∞ θ (R) = 0, such that, for R > |u| + ` and T ≥ α`2
¶
µ
Z
R − (|u| + `)
T
p/2
p
√
Wx0 [|LB(u,`) | ]dx0 ≤ C` `T exp −
T
|x0 |≥R
while for T ≤ αε2
Z
|x0 |≥R
Wx0 [|LTB(u,ε) |p/2 ]dx0
≤ CT
µ
R − (|u| + ε)
√
ε exp −
T
p/2 3
¶
.
Proof. Arguing as in the previous lemma, it is sufficient to prove that
µ
¶
Z
R − (|u| + `)
T
3
√
Wx0 [LB(u,`) ]dx0 ≤ C` T exp −
.
T
|x0 |≥R
We have
Z
|x0 |≥R
Wx0 [LTB(u,`) ]dx0
=
Z
dx0
B(0,R)c
3
= $` T
Z
T
0
Z
dt
T
T
dt
0
Z
Z
B(u,`)
≤ $`3 T g (R; T, u, `)
898
B(u,`)
dz pt (z − x0 )
dz
|B(u, `)|
Z
B(0,R)c
dx0 pt (z − x0 )
with (recall that ` ≤ 1)
g (R; T, u, `) =
sup
t∈(0,T ],|z|≤|u|+`
Z
B(0,R)c
dx0 pt (z − x0 ) .
Now (denoting by (Wt ) a 3D Brownian motion)
g (R; T, u, `) =
sup
t∈(0,T ],|z|≤|u|+`
P (|z − Wt | ≥ R)
≤ sup P (|Wt | ≥ R − (|u| + `))
t∈(0,T ]
µ
R − (|u| + `)
√
= sup P |W1 | ≥
t
t∈(0,T ]
µ
¶
R − (|u| + `)
√
≤ P |W1 | ≥
T
This completes the proof.
¶
¤
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